


Politecnico di Torino  
Academic Year 2016/17  
09BXTMQ Probability and statistics 

1st degree and Bachelorlevel of the Bologna process in Mathematics For Engineering  Torino 





Subject fundamentals
The course contents are divided into two parts, strictly interconnected: the first part is Probability, the second Mathematical Statistics.
In the first part, the main concepts of Measure Theory and Lebesgue Integration will be introduced, with an eye towards Probability Calculus. This way, the main mathematical results of Probability can be rigorously proved and more advanced topics can then be presented. The second part of the course is a review of the main statistical techniques based on Probability theory. The approach is less rigorous since applications will also be outlined. 
Expected learning outcomes
The student will learn the foundations of Probability Theory and Statistics from a rigorous mathematical standpoint and will learn methods and principles used to deal with uncertainty and variability in modern research.

Prerequisites / Assumed knowledge
Mathematical Analysis I e II, Geometry and Elementary Probability Calculus (as in the course Mathematical Methods in Engineering).

Contents
Elements of Measure Theory.
Lebesgue integration. Elementary Probability Calculus and random variables from a measuretheoretic point of view. Trasformations of random variables and random vectors. Sums of random variables. Characteristic functions. Conditional distributions. Conditional expected values. Convergence of sequences of random variables. Laws of large numbers. Central limit theorem and delta method. Order statistics. Multivariate normal and other distributions. Multivariate distributions via graphical methods. Probabilistic simulation. Sampling distributions. Asymptotic methods. Properties of statistical models and estimators. Fundamentals of point and interval estimation: coverage probability. Fundamentals of hypothesis testing: errors, power. Introduction to linear models: the full rank case. 
Delivery modes
Traditional lectures will be completed by exercise sessions and software sessions (either in labs or in class).

Texts, readings, handouts and other learning resources
P. Cannarsa, T. D'Aprile  Introduzione alla teoria della misura e all'analisi funzionale – Springer Verlag (2008)
G.B. Folland  Real Analysis: Modern Techniques and their Applications  John Wiley & Sons (1999) J. Jacod, P. Protter  Probability Essentials  Springer Verlag (2004) P.Baldi  Calcolo delle Probabilità  McGrawHill (2011) S.M. Ross – Calcolo delle Probabilità  Apogeo (2013) M. Gasparini  Modelli probabilistici e statistici  CLUT (2014) G. Casella, R.L. Berger  Statistical Inference  Duxbury Press (2002) 
Assessment and grading criteria
The exam is written and oral. The student will be tested on the ability to model simple problems dominated by uncertainty, using rigorous mathematical techniques.
A twohour written exam will cover all topics in the syllabus except Measure Theory. The assigned problems may also have a theoretical character and will require personal elaboration and a deep understanding of the topics seen in class. The exam is open book and open notes and will be worth at most 32 points. With a score higher than or equal to 15 the student is admitted to the oral exam. The oral exam will take place a few days after the written exam and will cover the entire syllabus. The student will have to demonstrate knowledge of the concepts and of the results seen in class, possibly with proof of results. The student will have to be able to provide examples and to solve simple exercises which may be asked by the teacher. The evaluation of the written and the oral exam together will contribute to the final mark. 
